Question: Carbon- $14$ is an element which loses $10\%$ of its mass every $871$ years. The mass of a sample of carbon- $14$ can be modeled by a function, $M$, which depends on its age, $t$ (in years). We measure that the initial mass of a sample of carbon- $14$ is $960$ grams. Write a function that models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. $M(t) = $
Answer: The strategy We can model the situation with an exponential function of the general form A ⋅ B f ( t ) A\cdot B\^{ f(t)}, where $A$ is the initial quantity, $B$ is a factor by which the quantity is multiplied over constant time intervals, and $f(t)$ is an expression in terms of $t$ that determines those time intervals. Let's use the given information to determine $A$, $B$, and $f(t)$. Understanding what's given We are given that the initial mass of the sample is $960$ grams, and that the sample loses $10\%$ of its mass every $871$ years. Note that losing $10\%$ is the same as being multiplied by $0.9$. [Why?] This means that the initial quantity is $A=960$ grams and the factor is $B=0.9$. We need to find $f(t)$ based on the fact that the quantity is multiplied by $0.9$ every $871$ years. Finding the expression in the exponent We know that the mass of the sample is multiplied by $0.9$ every $871$ years. This means that each time $t$ increases by $871$, $f(t)$ increases by $1$. Therefore, $f(t)$ is a linear function whose slope is $\dfrac{1}{871}$. When the initial measurement is made, the sample has all of its mass remaining. So $M(0) = 960$, which means that $f(0)=0$. [Why?] Therefore, $f(t)$ must be $\dfrac{t}{871}$. Summary We found that the following function models the mass of the carbon- $14$ sample remaining $t$ years since the initial measurement. M ( t ) = 960 ⋅ ( 0.9 ) t 871 M(t)=960\cdot (0.9)\^{ \frac{t}{871}}